What is a signal?

• A signal is a function that conveys information by mapping an independent variable (or variables) to measurable quantities.

• Formally, a signal is a mapping \(x:\ \mathcal{T}\rightarrow\mathcal{A}\), where \(\mathcal{T}\) is the domain (e.g., time \(t\in\mathbb{R}\) or sample index \(n\in\mathbb{Z}\)) and \(\mathcal{A}\) is the codomain (e.g., amplitudes in \(\mathbb{R}\) or \(\mathbb{C}\), vectors, or matrices).

Signal Classification

Signal Classification – Bounded

Signal Classification – Compact Support

Signal Classification – Causal

Signal Classification - Deterministic vs Random Signals

• A deterministic signal \(x_d(t)\) is completely specified by an explicit rule; it produces the same waveform on every observation (e.g., \(x_d(t)=A\cos(2\pi f_0 t+\phi)\)).

• A random signal (stochastic process) \(X(t)\) is a family of random variables indexed by \(t\); each observation yields a different realization. It is characterized statistically by its mean \(\mu_X(t)\) and autocorrelation \(R_X(\tau)\).

Signal Classification - Even/Odd

Signal Classification

Where:

\[x_{even}(t) = \frac{x(t)+x(-t)}{2} \] \[x_{odd}(t) = \frac{x(t)-x(-t)}{2} \]

Example

Example

\[x_{\text{even}}(t) = \frac{x(t) + x(-t)}{2}\]

\[x_{\text{odd}}(t) = \frac{x(t) - x(-t)}{2}\]

\[x(-t) = e^{-t}\]

\[x_{\text{even}}(t) = \frac{e^t + e^{-t}}{2} = \cosh(t)\]

\[x_{\text{odd}}(t) = \frac{e^t - e^{-t}}{2} = \sinh(t)\]

\[x(t) = x_{\text{even}}(t) + x_{\text{odd}}(t)\]

\[e^t = \cosh(t) + \sinh(t)\] ​

Example

Signal Classification - Energy vs Power Signals

Definitions. For a signal \(x(t)\) (continuous-time, CT) or \(x[n]\) (discrete-time, DT):

• Energy: \(E=\int_{-\infty}^{\infty}\lvert x(t)\rvert^2,dt\) (CT), \(\quad E=\sum_{n=-\infty}^{\infty}\lvert x[n]\rvert^2\) (DT).

• Average power: \(P=\lim_{T\to\infty}\dfrac{1}{2T}\int_{-T}^{T}\lvert x(t)\rvert^2,dt\) (CT), \(\quad P=\lim_{N\to\infty}\dfrac{1}{2N+1}\sum_{n=-N}^{N}\lvert x[n]\rvert^2\) (DT).

• Energy signal: \(0<E<\infty\) and \(P=0\) (e.g., \(x_E(t)=e^{-a t}u(t)\), \(a>0\), with \(E=\tfrac{1}{2a}\)).

• Power signal: \(0<P<\infty\) and \(E=\infty\) (e.g., \(x_P(t)=\cos(2\pi f_0 t)\), with \(P=\tfrac{1}{2}\)).

Fundamental Definitions

In signal processing, the classification of a signal \(x(t)\) depends on its behavior over an infinite time interval.

Total Energy (\(E\)) \[E = \int_{-\infty}^{\infty} |x(t)|^2 dt\]

Average Power (\(P\)) \[P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt\]

Energy Signals

A signal is classified as an energy signal if and only if: \[0 < E < \infty\] This condition implies that the average power \(P = 0\).

• Typically non-periodic.
• Finite duration or rapidly decaying.

Example: Exponential Decay

Consider the signal \(x(t) = e^{-at} u(t)\) where \(a > 0\).

Proof of Energy: \[E = \int_{0}^{\infty} (e^{-at})^2 dt = \int_{0}^{\infty} e^{-2at} dt\]

Evaluating the definite integral: \[E = \left[ \frac{e^{-2at}}{-2a} \right]_{0}^{\infty} = 0 - \left( \frac{1}{-2a} \right) = \frac{1}{2a}\]

Conclusion: Since \(E\) is a finite value, it is an energy signal.

Power Signals

A signal is classified as a power signal if and only if: \[0 < P < \infty\] This condition implies that the total energy \(E = \infty\).

• Typically periodic.
• Infinite duration without decay.

Example: Sinusoidal Signal

Consider \(x(t) = A \cos(\omega_0 t)\).

Proof of Power: Using the period \(T = \frac{2\pi}{\omega_0}\): \[P = \frac{1}{T} \int_{0}^{T} |A \cos(\omega_0 t)|^2 dt = \frac{A^2}{T} \int_{0}^{T} \frac{1 + \cos(2\omega_0 t)}{2} dt\]

The integral of the oscillating term over a full period is zero: \[P = \frac{A^2}{2T} [t]_0^T = \frac{A^2}{2}\]

Conclusion: Power is finite (\(P = A^2/2\)), thus it is a power signal.

Summary Table

Python Example

Independent Variable Transformations

Time Scaling

• Definition: Changes the time scale of the signal. [ x(at), a > 1 , < a < 1 ]
• Example: If ( x(t) = (t) ), then ( x(2t) ) is compressed.

Time Shifting

• Definition: Shifts the signal in time. [ x(t - t_0) ]
• Example: ( x(t - 2) ) shifts the signal 2 units to the right.

Time Reversal

• Definition: Flips the signal across the vertical axis. [ x(-t) ]
• Example: If ( x(t) = t^2 ), then ( x(-t) = t^2 ) (even signal).

Dependent Variable Transformations

Amplitude Scaling

• Definition: Multiplies the amplitude by a scalar factor. [ a x(t), a > 1 , < a < 1 ]
• Example: If ( x(t) = (t) ), then ( 2x(t) ) doubles the amplitude.

Amplitude Shifting

• Definition: Adds a constant value to the amplitude. [ x(t) + c ]
• Example: If ( x(t) = (t) ), then ( x(t) + 2 ) shifts the signal up by 2 units.

Combined Transformations

Example

Consider: [ y(t) = 2 x(3t - 1) + 1 ] 1. Time compression: ( x(3t) ) compresses the signal. 2. Time shift: ( x(3t - 1) ) shifts it to the right by 1 unit. 3. Amplitude scaling: ( 2 x(3t - 1) ) amplifies the signal. 4. Amplitude shift: ( +1 ) shifts it upward.

Visualization Example in Python